Integers

The Asian School, Kingdom of Bahrain 2025 – 2026

Subject: Mathematics
Grade: 7
Chapter 1 INTEGERS
IMPORTANT POINTS

 Integers.
 Addition & Subtraction of Integers and its properties.
 Multiplication & Division of Integers and its properties.
Integers
All-Natural numbers, 0 and Negative of Counting numbers are called Integers.

 Zero (0) is an integer which is neither positive nor negative.

 There is no greatest or smallest integer.
 Zero (0) is greater than every Negative Integer and less than every Positive Integer.
 Every Positive Integer is greater than every Negative Integer.
(+ve) + (+ve) = (+ve)
(-ve) + ( -ve) = (-ve)
(+ve) + (-ve) = subtract and keep the
(-ve) + (+ve) = sign of the greater number

Properties of Addition of integers

 The sum of two integers is always an integer.

 For any two integers a and b, a+b is an Integers are closed under
Closure Property integer. addition.

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Commutative Property  Two integers can be added in any order. Addition is commutative
 If a and b are any integers, then for integers.
a+b = b+a.
 Three or more integers can be grouped in any
Associative property order to find their sum. Addition is associative for
 If a, b and c are any three integers, then integers.
(a+b) + c = a + (b+c)
 The sum of any integer and 0 is the integer
itself. ‘0’ is the additive identity
Additive Identity  If a is an integer, then for integer.
a+0=0+a=a
 The sum of an integer and its additive Integer a and (-a) are
Additive Inverse inverse is zero. additive Inverse of each
 If a is an integer, then a + (-a) = 0 other.

Properties of Subtraction of Integers

 The difference of two integers is always

Closure Property an integer. Integers are closed under
 If a and b are any two integers then a-b subtraction.
is always an integer.
Commutative property  If a and b are any two integers then Subtraction is not commutative
for integers.
a-b≠b-a
Associative property  If a, b and c are any three integers then Subtraction is not associative
(a – b) – c ≠ a – (b – c) for integers.

Exercise 1.1

Qn 1. Write down a pair of integers whose:

(a) sum is -7

Solution:- Let consider the integers (-5) and (-2)

Then (-5)+(-2) = (-7)

Therefore the pair is (-5, -2)

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(b) difference is – 10

Solution: -
Let consider the integers (-15) and (-5)
Then, (-15) – (-5)
= – 15 + 5 = -10
Therefore, the pair is (-15, -5)

(c) sum is 0
Solution:-
Let consider the integers 6 and (-6)
Then, 6 + (-6)
=6–6=0
Therefore, the pair is (6, -6)
2. (a) Write a pair of negative integers whose difference gives 8
Solution:-
Let consider the integers (-7) and (-15)
Then, (-7) – (- 15)
= (-7) + 15 = 8
Hence the pair is (-7, -15)
(b) Write a negative integer and a positive integer whose sum is – 5.
Solution:-

Let consider the integers (-17) and 12.

Then (-17) + 12 = -5
Hence, the integers are -17 and 12
(c) Write a negative integer and a positive integer whose difference is – 3.
Solution:-
Let consider the integers (-2) and 1
Then, (-2) – (1) = (-2) +(-1) = – 3
Hence, the integers are (-2) and 1.

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Qn 3. In a quiz, team A scored – 40, 10, 0 and team B scored 10, 0, – 40 in three successive rounds.
Which team scored more? Can we say that we can add integers in any order?

Solution

Score of team A = -40, 10, 0

Total score obtained by team A = – 40 + 10 + 0 = – 30 Score
of team B = 10, 0, -40
Total score obtained by team B = 10 + 0 + (-40)
= 10 + 0 – 40 = – 30
Thus, the score of both A team and B team is same. Yes, we
can add integers in any order.

H.W

I. Multiplication of Integers
 The product of two integers with like signs is positive.
 The product of two integers with unlike signs is negative.
(+ve) × (+ve) (+ve) × (-ve)
(-ve) × (-ve) (+ve) (-ve) × (+ve) (-ve)
Like signs Unlike signs

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Properties of Multiplication of Integers

 The product of any two integers is an integer.

Closure Property  If a and b are any two integers, then Integers are closed under
a×b = integer. multiplication.
 Two integers can be multiplied in any order.
Commutative  If a and b are any two integers, then Multiplication is commutative
property a×b=b×a for integers.
 Three or more integers can be grouped in any
Associative Property order to find their product. Multiplication is associative
 If a, b and c are any three integers, then for integer.
(a×b)×c = a × (b×c)
 The product of any integer and 1 is the 1 is the multiplicative identity
Multiplicative integer itself. for integers.
Identity  If a is an integer then, a×1 = 1×a
 The product of any integer with zero is zero.
Property of zero  If a is an integer then a×0 = 0×a =0
Distributive property  If a, b and c are three integers, then
of Multiplication a×(b+c) = (a×b) + (a×c)
over addition.  Eg: 3×(4+5) = (3×4) + (3×5)
= 12 + 15 = 27
Distributive property  If a, b and c are three integers, then
of Multiplication a×(b - c) = (a×b) – (a×c)
over subtraction.  Eg: 2×(5-9) = (2×5) – (2×9)
= 10 – 18 = -8

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Exercise 1.2

Qn 1. Find each of the following products:

(a) 3 × (–1)
Solution:-
3 × (-1) = -3
(c) (–21) × (–30) (+) × (-) = -
Solution:-
(-21) × (-30) = 630
(-) × (–) = +
(g) 9 × (–3) × (– 6)
Solution:-
9 × (-3) × (-6) = 9 × 18 = 162 (-) × (–) = +
(i) (–1) × (–2) × (–3) × 4
Solution:-
(-1) × (-2) × (-3) × (4) = [(–1) × (–2)] × [(–3) × 4]
= 2 × (-12) = – 24
[(-) × (–) = +], [(-) × (+) = -]

Qn. 2 Verify the following:

(b) (–21) × [(– 4) + (– 6)] = [(–21) × (– 4)] + [(–21) × (– 6)]

Solution:-
From the given equation,
LHS RHS
(–21) × [(– 4) + (– 6)] [(–21) × (– 4)] + [(–21) × (– 6)]
= (-21) × [-4 – 6] = [84] + [126]
= (-21) × [-10] = 210 = 210

210=210 LHS=RHS
,VERIFIED

HW Q 2 (a), Q 3

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Division of Integers
 The quotient of two integers with like signs is positive.
 The quotient of two integers with unlike signs is negative.

(+ve) ÷ (+ve) (+ve) ÷ (-ve)

(-ve) ÷ (-ve) (+ve) (-ve) ÷ (+ve) (-ve)
Like signs Unlike signs
Properties of Division

Integers

i. Are not closed under Division. [a ÷ b is not always an integer]

ii. Does not hold Commutative property. [ a ÷ b ≠ b ÷ a ]
iii. Does not hold Associative property. [ (a ÷ b) ÷ c ≠ a ÷ (b ÷ c)]
iv. No identity element for division. [a ÷ 1 = a; 1 ÷ a ≠ a ie, a ÷ 1 ≠ 1 ÷ a]

v. Any integer divided by 1 gives the same integer. [a ÷ 1 = a]

vi. Any integer divided by (-1) gives negative of the integer. [a ÷ (-1) = -a]
vii. For any integer a, a ÷ 0 is not defined but 0 ÷ a = 0 for a ≠ 0

Exercise 1.3

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Q 1. Evaluate each of the following:

(a) (–30) ÷ 10 (c) (–36) ÷ (–9)

Solution:- Solution:-
(–30) ÷ 10 = – 3 (-36) ÷ (-9) = 4
(i) [(– 6) + 5] ÷ [(–2) + 1] HW Q 1 (b), (d), (e), (f), (g), (h)
Solution:-
[(-6) + 5] ÷ [(-2) + 1]
= [-1] ÷ [-1] = 1

Q 2. Verify that a ÷ (b + c) ≠ (a ÷ b) + (a ÷ c) for each of the following values of a, b and c.

(a) a = 12, b = – 4, c = 2
Solution:-
To verify, a ÷ (b + c) ≠ (a ÷ b) + (a ÷ c)
Given, a = 12, b = – 4, c = 2
LHS a ÷ (b + c) RHS (a ÷ b) + (a ÷ c)
= 12 ÷ (-4 + 2) = (12 ÷ (-4)) + (12 ÷ 2)
= ……12÷( -2). = ……-3 + 6……. HW Q 2 (b), 3, 4
= ………(-6)…. = ……3……..

By comparing LHS and RHS, -6 ≠ 3

Thus LHS ≠ RHS; a ÷ (b + c) ≠ (a ÷ b) + (a ÷ c) Hence, the given values are verified

Q 3. Fill in the blanks:

(a) 369 ÷ __1___ = 369

(b) (–75) ÷ __75___ = –1
(c) (–206) ÷ __(-206)___ = 1
(d) – 87 ÷ ____(-1)_ = 87
(e) _(-87)____ ÷ 1 = – 87
(f) ___(-48)__ ÷ 48 = –1
(g) 20 ÷ _(-10)____ = -2
(h) _(-12)____ ÷ (4) = –3

Q4. Write five pairs of integers (a, b) such that a ÷ b = –3. One such pair is (6, –2) because

6 ÷ (– 2) = (–3).

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Solution:-

(i) (15, -5)

Because, 15 ÷ (–5) = (–3)

(ii) (-15, 5)

Because, (-15) ÷ (5) = (–3)

(iii) (18, -6)

Because, 18 ÷ (–6) = (–3)

(iv) (-18, 6)

Because, (-18) ÷ 6 = (–3)

(v) (21, -7)

Because, 21 ÷ (–7) = (–3)

Q 5. The temperature at 12 noon was 10oC above zero. If it decreases at the rate of 2oC per hour
until midnight, at what time would the temperature be 8°C below zero? What would be the temperature
at midnight?

Solution:-

From the question, given that,

The temperature at the beginning, i.e. at 12 noon = 10oC

Rate of change of temperature = – 2oC per hour

Then,

Temperature at 1 p.m. = 10 -2= 8oC

Temperature at 2 p.m. = 8 – 2 = 6oC

Temperature at 3 p.m. = 6 – 2 = 4oC

Temperature at 4 p.m. = 4 – 2 = 2oC

Temperature at 5 p.m. = 2 – 2 = 0oC

Temperature at 6 p.m. = 0 – 2 = -2oC

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Temperature at 7 p.m. = -2 -2 = -4oC

Temperature at 8 p.m. = -4 – 2 = -6oC

Temperature at 9 p.m. = -6 – 2 = -8oC

Temperature at 10 p.m = -8-2 = -10 oC

Temperature at 11p.m = -10 -2 = -12 oC

Temperature at 9 p.m = -12 – 2 = -14 oC

At 9pm the temperature would be 8 oC below zero and the temperature would be -14 oC at midnight.

Q 6. In a class test (+ 3) marks are given for every correct answer and (–2) marks are given for every
incorrect answer and no marks for not attempting any question. (i) Radhika scored 20 marks. If she
has got 12 correct answers, how many questions has she attempted incorrectly? (ii) Mohini scores –5
marks in this test, though she has got 7 correct answers. How many questions has she attempted
incorrectly?

Solution
Marks awarded for 1 correct answer = + 3
Marks awarded for 1 wrong answer = -2
(i) Radhika scored 20 marks
Total marks awarded for 12 correct answers = 12 × 3 = 36
Marks awarded for incorrect answers = Total score – Total marks awarded for 12 correct answers
= 20 – 36 = …-16……..
So, the number of incorrect answers made by Radhika = (-16) ÷ (-2) = …8….
(ii) Mohini scored -5 marks
Total marks awarded for 7 correct answers = 7 × 3 = 21
Marks awarded for incorrect answers = Total score – Total marks awarded for 7 correct answers
= – 5 – 21 = ……-26……
So, the number of incorrect answers made by Radhika = …-26… ÷. -2…. = …13….

Q7. An elevator descends into a mine shaft at the rate of 6 m/min. If the descent starts from 10 m above
the ground level, how long will it take to reach – 350 m?

Solution: -

From the question,

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The distance at which the elevator starts = 10 m

The final depth of the elevator = – 350 m … [∵distance descended is denoted by a negative

integer]

The distance between 10m and (-350) m= 10 – (-350) = 10+350

= 360 m

Then,

Time taken by the elevator to descend 6 m = 1 min

Time = Distance ÷Speed

= 360÷ 6

= 60 minute =1 hour

Therefore, the elevator will take 1 hour to reach at the distance of (-350) m

Reference Link: https://www.learncbse.in/integers-class-7-extra-questions/

: https://youtu.be/u3HANGB47pA

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